I am reading Hiroshi Kunita's lecture Stochastic Differential Equations Based on Levy Processes and Stochastic Flows of Diffeomorphisms.It proves Ito's formula for cadlag semimartingale.
The proof is concise, but I doubt that it is not correct. Right in this picture, it constructs a function r(x,y)(in the first highlight), and claims that it suffices the identity(in the second highlight), but I just found the identidy is not valid since we can not get F by the integraton of the second order derivative F'' (this will only get F' instead of F). So I think this is proof is incorrect. Did I make some mistake?
Your doubt is justified, the Taylor polynomial with integral remainder term is $$ F(y)=F(x)+F'(x)(y-x)+\frac12\int_0^1(1-\theta)F''(x+\theta(y-x))(y-x)^2\,d\theta. $$ As $|1-\theta|\le 1$ on the unit interval, this omission likely has no consequences for the proof. It is reasonable to suspect that this is a late transmission error from a manuscript to the print version.